In certain applications it is very beneficial to be able to predict the future velocity of a moving object. For example, such information could be employed to calculate velocities in relation to automobile air-bag use. This information could then be employed to more quickly deploy air-bags and/or to control their rate of deployment. Other instances of where prediction of future velocity values may be implemented is in controlling the operation of the moving object, in regulating the spacing of moving vehicles on an automated highway system and the like.
Past integrators and related methods for predicting velocity from accelerometer samples have been polynomial based and have been documented as not providing accurate predictions. A skilled artisan is aware that a general explicit linear multi-step integration method can be written as ##EQU1##
where T is the uniform sampling interval. Traditional methods usually assume knowledge of only one function value, and that is generally chosen as the most recent value, to give a formula of the form ##EQU2##
Typically, samples of acceleration are integrated to obtain a velocity value. Thus, samples of f'(t) represent acceleration samples and f(t-T) represents a corresponding velocity value.
When the coefficients in formulas similar to Equation (2) are computed so that the formula is accurate for the highest degree polynomials possible, then one obtains the well-known Adams-Bashforth (AB) method. For example, N=1 gives the Euler formula EQU .function.(t)=.function.(t-T)+T.function.'(t-T), (3)
or with N=2 one obtains the AB two-step method, ##EQU3##
The three-step AB formula is ##EQU4##
valid for polynomials of the third degree. Such formulas can be derived by using Lambert's equations and by using matrix methods to find the coefficients in the formula. These AB methods are explicit methods, since they are based on past samples of the function and derivative.
Predictor-corrector integration methods employ both a predictor, which is an explicit type of formula as given above, as well as a corrector, which is an implicit formula that includes a current value of the derivative, i.e., f'(t). The Adams-Moulton (AM) family of formulas is the usual companion to the AB family of equations described above and is similarly polynomial-based. The coefficients in the corrector formula, ##EQU5##
are chosen so that the formula is accurate for polynomials of the highest degree possible. The AM-one (trapezoidal) and AM-two step formulas are given, respectively, by
.function.(t)=.function.(t-T)+T[1/2.function.'(t)+1/2.function.'(t-T)] (7) EQU .function.(t)=.function.(t-T)+T[5/12.function.'(t)+8/12.function.'(t-T)-1/ 12.function.'(t-2T)] (8)
Any method utilizing the above equations for predicting velocity or for determining any appropriate function from knowledge of its derivative's values are based upon the polynomial and have not been established as providing the most accurate predictions thereof.